Find the Exact Value sin(arctan(- square root of 3))

Problem

sin(arctan(−√(,3)))

Solution

1. Identify the inner value of the inverse tangent function. We need to find an angle θ such that tan(θ)=−√(,3) within the restricted range of the arctangent function, which is (−π/2,π/2)

2. Determine the angle θ Since tan(π/3)=√(,3) and the tangent function is odd, we have tan(−π/3)=−√(,3) Thus, θ=−π/3

3. Substitute the angle back into the sine function. The expression becomes sin(−π/3)

4. Evaluate the sine of the angle. Using the unit circle or trigonometric properties, sin(−π/3)=−sin(π/3)=−√(,3)/2

Final Answer

sin(arctan(−√(,3)))=−√(,3)/2

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